This is from an early version of Chapter 7 and from my 2009 presentation on avatars at the conference Two Streams in the Philosophy of Mathematics organized by David Corfield and Brendan Larvor. Some of the first half has been preserved on p. 215 and there is a hint of the remainder on p. 201. It reads well enough but didn’t make the final cut. The last sentence needs further elaboration.

What does the word *motive* mean? I’m not sure I know, although a provisional definition is available. The danger identified by Ivor Grattan-Guinness as “the royal road to me”-type history consists in projecting present notions abusively on historical texts, as illustrated in several places in André Weil’s *History of Mathematics: How and Why?*:

For reasons such as this the algebraic reading of Euclid has been discredited by specialist historians in recent decades. By contrast, it is still advocated by mathematicians, such as Weil … who even claimed that group theory is necessary to understand Book 5 (introducing ratios, and forming propositions and other theorems involving geometrical magnitudes) and Book 7 (introducing basic properties of positive integers)! [1]

The “royal road to me” is abusive retrospection. On the other hand, from the historian’s perspective, the *avatar* is a (possibly abusive) projection of a speculative future mathematics on current practice, and it would be odd to do history without any consideration for the past. But the mathematician’s understanding is not directed to history but rather to other mathematicians.

As a vehicle for communication among mathematicians, is the avatar ontological, epistemological, or methodological? For example, by affirming that all elliptic curves are modular, is one saying (as Weil might) that when we are talking about elliptic curves over **Q** we are “really” talking about modular elliptic curves? Or is it rather that, as working number theorists (myself included) are happy to recognize, that when we want to find out something about an elliptic curve over **Q**, we are entitled to use the now established theorem that it is (also?) a modular elliptic curve, and that therefore the way to understand an elliptic curve over any number field is as whatever is the appropriate analogue of modularity for that field, which is provided by conjectures of Langlands that in general are far from being proved? I’m led to ask this question by this quotation from a book entitled *Relevance*:[2]

Human spontaneous non-demonstrative inference is not, overall, a logical process. Hypothesis formation involves the use of deductive rules, but is not totally governed by them; hypothesis confirmation is a non-logical cognitive phenomenon: it is a by-product of the way assumptions are processed, deductively or otherwise.

This passage would be relevant to avatars if one of their functions were to contribute to a collective process of formation of mathematical hypotheses, namely about the ulterior objects of which the accessible ones are presumed to be avatars, which seems unquestionable; and if normal mathematical reasoning with the avatars, which is possible precisely because they are accessible, contributed to an ultimate process of confirmation of these hypotheses, which is one plausible reason for their use in practice. Grothendieck’s hypothesis that known commonalities among cohomological theories of algebraic varieties can be traced back to “common motives” is consistent with logic but is not exhausted by its logical features, and while I’m not sure how the end of the passage might apply to the eventual confirmation of this hypothesis, it’s justified by the likelihood that it will involve constructions that are not directly deduced from the terms in which the hypothesis is framed.

[1] I. Grattan-Guinness, The mathematics of the past: distinguishing its history from our heritage, *Historia Mathematica*, Volume 31, Issue 2, May 2004, Pages 163-185.

[2] by Dan Sperber and Deirdre Wilson, Blackwell, second edition 1995, p. 69.

Jon AwbreyFrom a Peircean standpoint you have just raised the question of abductive reasoning and its role in the cycle of inquiry.

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mathematicswithoutapologiesPost authorPeirce is mentioned in the Sperber-Wilson book in connection with semiotics but not with reasoning.

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Jon AwbreyAs I read him, Peirce began with a quest to understand how science works, which required him to examine how symbolic mediations inform inquiry, which in turn required him to develop the logic of relatives beyond its bare beginnings in De Morgan. There are therefore intimate links, which I am still trying to understand, among his respective theories of inquiry, signs, and relations.

There’s a bit on the relation between interpretation and inquiry here and a bit more on the three types of inference — abduction, deduction, induction — here.

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sntxFrankly, sometimes even demonstrative inferences trouble me. The reason is somewhat captured by Lewis Carroll’s paradox of Achilles and the Tortoise. Perhaps one could give a sort of ‘Kripkensteinian’ explanation to the paradox but I’m not sure.

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